A matrix is a rectangular collection of numbers (again, possibly complex).
It might represent a collection of row/column vectors, or a transformation
that changes a vector into a different vector. For example, the rotation
of a real 2D vector by an angle about the axis could be
represented as a 2x2 matrix,

The effect of this rotation on a 2D vector in the plane can
be determined by multiplying the vector by this transformation matrix,

(9)

where
and
.

More generally, the product of a matrix and a column vector may be
written as

(10)

where

(11)

Notice that each element of the product, , is just the
dot product of the th row of the matrix with the vector.
It is also possible to think of the column vector as
being a linear combination of the columns of the matrix ,
with each column having a weight .

It is also possible to ``left-multiply'' a matrix by a row vector, like
this:

(12)

where

(13)

This time, each element of the product, , is the dot product
of the vector on the left with the th column of the matrix.
The row vector may be thought of as a linear combination
of the rows of the matrix , with each row having a weight
of .

From the information given, it should be obvious that a row vector
times a matrix times a column vector yields a number (scalar).

Now consider the multiplication of two matrices and ,
which yields a new matrix, :

(14)

where

(15)

That is, each element of the product matrix
is the result of a dot product between
row of the matrix and column of the matrix .
It is possible to think of matrix multiplication as a generalization
of the product of a matrix times a vector, where now instead of one
column vector on the right, we have a series of them. It is also
possible to think of matrix multiplication as a generalization of
the product of a row vector times a matrix, where now instead of
one row vector on the left, we have a series of them.

The transpose of a matrix , typically denoted
, is obtained simply by swapping elements across the diagonal,
. For example,

(16)

For complex numbers, usually the complex conjugate transpose, or
adjoint, is more useful. The adjoint of a matrix is often
denoted
. For example,

(17)

If a matrix is equal to its adjoint
, it is
said to be a Hermitian matrix. Clearly, this can only happen if
the diagonal elements are real (otherwise, will never equal ,
for example). One can also take the adjoint of a vector, which is
effectively what one does in order to take a dot product of two complex
vectors (see above):